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The paper "The Seven Virtues of Simple Type Theory" mentions that it uses the same trick (due to Tarski) to define the semantics that is also used by first-order logic. I interpreted this a reference to Tarski's Truth Definitions. After reading the linked SEP entry, I learned that there is a version from 1933 and a version from 1956.

If I understood it correctly, for the 1933 version, the model (i.e. the algebraic structure about which we talk) is part of the meta-language and not mentioned separately. The assignment of objects to variables on the other hand is what can satisfy a given formula. A formula is (defined to be) true if it is satisfied by all possible assignments of objects to variables.

The 1956 version is treated less explicitly in the linked SEP entry, but it is hinted at that the model is no longer an implicit part of the meta-language, but an explicit object from set-theory. A model can satisfy a given formula (or sentence), similar to how an "assignment of objects to variables" could satisfy a given formula for the 1933 version. But the text also hints that the 1956 now relies stronger on an underlying set-theory, while the 1933 explicitly tried to minimize "the set-theoretic requirements of the truth definition".

Are these all the important differences between the 1933 and the 1956 version? Are there any important misunderstandings in my summary of the differences?

  • I don't think there are any important misunderstandings (though I'm not sure there was any talk of models, in the model theoretic sense, in the 1933 paper). I'm curious though so ill take a look in a few days when I return from vacation. – Dennis Aug 16 '13 at 10:13
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    Dennis, I think you're right; Tarski was still thinking of truth as satisfaction by "all objects" if memory serves. You can see where this kind of truth definition is going to run into trouble with abstract model theory. – Paul Ross Aug 21 '13 at 0:08
  • @PaulRoss I understood the SEP entry in the way that "the model was not mentioned separately", in order to minimize "the set-theoretic requirements". Given that Tarski was familiar with Löwenheim's theorem from Löwenheim's publication "Über Möglichkeiten im Relativkalkül" (1915), I assume that Tarski intentionally avoided to talk about models (in a set-theoretic sense). (And yes, I have checked that Tarski's publication indeed included Löwenheim's publication in the bibliography.) – Thomas Klimpel Aug 21 '13 at 1:16
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The issue is a little bit complex.

See :

John Etchemendy, TARSKI ON TRUTH AND LOGICAL CONSEQUENCE, in The Journal of Symbolic Logic, Vol.53 (1988)

Mario Gomez-Torrente, Tarski on Logical Consequence, in ND Journal of Formal Logic, Vol.37 (1996)

Craig Bach, TARSKI'S 1936 ACCOUNT OF LOGICAL CONSEQUENCE, in Modern Logic, Vol.7 (1997)

the new translation of Alfred Tarski, On the Concept of Following Logically (1936), in HISTORY AND PHILOSOPHY OF LOGIC, 23 (2002).

I'll try to summarize the central issue, trying to avoid technical details (due also to the lack of LaTeX editor).

Reference is, of course, to Wilfrid Hodges' entry in SEP : Tarski's Truth Definitions.

We say that a language is fully interpreted if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted [...]. This was the main difference between the 1933 definition and the later model-theoretic definition of 1956 [...].

In 1933 Tarski assumed that the formal languages that he was dealing with had two kinds of symbol, namely constants and variables. The constants included logical constants, but also any other terms of fixed meaning. The variables had no independent meaning and were simply part of the apparatus of quantification.

Today logicians work with formal languages, whose extra-logical constants are uninterpreted (until an interpretation is fixed in a particular case). To provide an interpretation or model of such a language, one typically specifies a domain or universe of discourse and assigns to each individual constant a unique object in the domain, to each monadic first-order predicate a subset of the domain, and so on.

Tarski [1936] worked instead with what he called formalized languages, in which the extra-logical constants are interpreted and the domain is fixed.

As Hodges explains :

Model theory by contrast works with three levels of symbol. There are the logical constants (=, ¬, & for example), the variables (as before), and between these a middle group of symbols which have no fixed meaning but get a meaning through being applied to a particular structure. The symbols of this middle group include the nonlogical constants of the language, such as relation symbols [like ∈ in set theory], function symbols and constant individual symbols [like 0 in arithmetic]. They also include the quantifier symbols ∀ and ∃, since we need to refer to the structure to see what set they range over. This type of three-level language corresponds to mathematical usage; for example we write the addition operation of an abelian group as +, and this symbol stands for different functions in different groups.

By the late 1940s it had become clear that a direct model-theoretic truth definition was needed. The version we use today is based on that published by Tarski and Robert Vaught in 1956.

The right way to think of the model-theoretic definition is that we have sentences whose truth value varies according to the situation where they are used. So the nonlogical constants are not variables; they are definite descriptions whose reference depends on the context. Likewise the quantifiers have this indexical feature, that the domain over which they range depends on the context of use.

Because Tarski [1936] uses a formalized language with interpreted extra-logical constants, he must first replace extralogical constants by variables of the same type and then consider what sets of objects satisfy the resulting sentential function.

The concept of a model in Tarski [1936] is not the contemporary concept. In contemporary mathematics, as Hodges points out, a model or structure is roughly a collection of elements with relations defined on them. A set of objects is not such a structure.

  • Good, but to me this raises the question of how did Tarski think about "interpreted languages". Because if he simply thought that interpretation=extension, then how would his 1933 view substantially differ from the modern way of understanding interpretation as giving extensions for non-logical symbols and a domain for the variables. If interpretation is extension then varying the interpretation of a language should not be a conceptual problem at all. – Johannes Jun 30 '15 at 1:01
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    @Johannes - you can see John Etchemendy's article (1988) cited above : page 68-on, dedicated to a discussion of Tarski's account. – Mauro ALLEGRANZA Jun 30 '15 at 12:01

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